A vertical compression is "squishing" all points towards the x-axis, it is also important to rember that the invariant points in such a transformation are ON the x-axis. A horizontal compression is "squishing" all points towards the y-axis. It is important to remember that all invariant points are on the y-axis. However, there are some cases where a transformation is equivalent to another transformation relative to a different axis.
Consider the function [tex]f(x) = 4x^2 [/tex] This is a transformation of the graph of [tex]y = x^2[/tex] vertically stretched by a factor of 2. However, this is also equivalent to a horizontal compression by a factor of 1/2.